The Sequence of Lucas Numbers Is Not Stable modulo 2 and 5

Let L0 = 2, L1 = 1, and Ln = Ln−1+Ln−2 for n ≥ 2, denote the sequence L of Lucas numbers. For any modulus m ≥ 2, and residue b (mod m), denote by vL(m, b) the number of occurrences of b as a residue in one (shortest) period of L modulo m. In this paper, we completely describe the functions vL(p , .) for k ≥ 1 in the cases p = 2 and p = 5. Using a notion formally introduced by Carlip and Jacobson, our main results imply that L is neither stable modulo 2 nor modulo 5. This strikingly contrasts with the known stability of the classical Fibonacci sequence modulo these two primes. Communicated by Vera T. Sós Dedicated to the memory of Professor Edmund Hlawka